Semiregular Tessellation — Definition, Formula & Examples
A semiregular tessellation is a tiling of the plane using two or more types of regular polygons, arranged so that the same pattern of polygons meets at every vertex.
A semiregular (or Archimedean) tessellation is an edge-to-edge tiling of the Euclidean plane by two or more distinct species of regular polygons such that every vertex figure is congruent — that is, the cyclic sequence of polygons surrounding each vertex is identical up to reflection. There are exactly eight such tilings.
How It Works
To identify or construct a semiregular tessellation, you check that every polygon is regular (all sides and angles equal) and that the arrangement of polygons around each vertex is the same throughout the tiling. Each tiling is described by its vertex configuration — a list of the polygon sizes in order around a vertex. For example, 3.6.3.6 means a triangle, hexagon, triangle, hexagon meet at every vertex. The interior angles at each vertex must sum to exactly for the polygons to fit together without gaps or overlaps.
Worked Example
Problem: Verify that regular triangles and regular hexagons can form the semiregular tessellation with vertex configuration 3.6.3.6.
Step 1: Find the interior angle of each polygon. For a regular -gon, the interior angle is .
Step 2: The vertex configuration 3.6.3.6 places two triangles and two hexagons at each vertex. Sum their interior angles.
Step 3: Since the angles sum to exactly 360°, these polygons fit perfectly around every vertex with no gaps or overlaps.
Answer: The angle sum is , confirming that 3.6.3.6 (the trihexagonal tiling) is a valid semiregular tessellation.
Why It Matters
Semiregular tessellations appear in architectural design, decorative tilework, and materials science (such as graphene's honeycomb lattice). Understanding vertex configurations also builds the reasoning you need for studying Archimedean solids, which are the 3D analogs of these tilings.
Common Mistakes
Mistake: Confusing semiregular tessellations with regular tessellations.
Correction: Regular tessellations use only one type of regular polygon (there are exactly three: triangles, squares, hexagons). Semiregular tessellations require two or more types of regular polygons while keeping every vertex identical.